A rotary part of a crane is associated with an electric slewing mechanism including at least one geared electric motor unit of which the electric motor is powered in such a way as to produce a slewing torque having a certain maximum value. When the crane is in use and for conditions such as windspeed greater than a given value, jib length greater than a given value and moment of the lifted load greater than a given value, the maximum value of a slewing torque produced by the electric motor is increased. The crane thus becomes easier to drive, particularly in the phases during which the jib is adopting a “windward trend”.